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In mathematical biology, the community matrix is the linearization of a generalized Lotka–Volterra equation at an equilibrium point.^[1] The eigenvalues of the community matrix determine the stability of the equilibrium point.

For example, the Lotka–Volterra predator–prey model is

{\begin{array}{rcl}{\dfrac {dx}{dt}}&=&x(\alpha -\beta y)\\{\dfrac {dy}{dt}}&=&-y(\gamma -\delta x),\end{array}}

where x(t) denotes the number of prey, y(t) the number of predators, and α, β, γ and δ are constants. By the Hartman–Grobman theorem the non-linear system is topologically equivalent to a linearization of the system about an equilibrium point (x*, y*), which has the form

{\begin{bmatrix}{\frac {du}{dt}}\\{\frac {dv}{dt}}\end{bmatrix}}=\mathbf {A} {\begin{bmatrix}u\\v\end{bmatrix}},

where u = x − x* and v = y − y*. In mathematical biology, the Jacobian matrix $\mathbf {A}$ evaluated at the equilibrium point (x*, y*) is called the community matrix.^[2] By the stable manifold theorem, if one or both eigenvalues of $\mathbf {A}$ have positive real part then the equilibrium is unstable, but if all eigenvalues have negative real part then it is stable.

References

^ Berlow, E. L.; Neutel, A.-M.; Cohen, J. E.; De Ruiter, P. C.; Ebenman, B.; Emmerson, M.; Fox, J. W.; Jansen, V. A. A.; Jones, J. I.; Kokkoris, G. D.; Logofet, D. O.; McKane, A. J.; Montoya, J. M; Petchey, O. (2004). "Interaction Strengths in Food Webs: Issues and Opportunities". Journal of Animal Ecology. 73 (5): 585–598. doi:10.1111/j.0021-8790.2004.00833.x. JSTOR 3505669.
^ Kot, Mark (2001). Elements of Mathematical Ecology. Cambridge University Press. p. 144. ISBN 0-521-00150-1.

Murray, James D. (2002), Mathematical Biology I. An Introduction, Interdisciplinary Applied Mathematics, vol. 17 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95223-9.

This applied mathematics–related article is a stub. You can help Wikipedia by expanding it.

[1] Berlow, E. L.; Neutel, A.-M.; Cohen, J. E.; De Ruiter, P. C.; Ebenman, B.; Emmerson, M.; Fox, J. W.; Jansen, V. A. A.; Jones, J. I.; Kokkoris, G. D.; Logofet, D. O.; McKane, A. J.; Montoya, J. M; Petchey, O. (2004). "Interaction Strengths in Food Webs: Issues and Opportunities". Journal of Animal Ecology. 73 (5): 585–598. doi:10.1111/j.0021-8790.2004.00833.x. JSTOR 3505669.

[2] Kot, Mark (2001). Elements of Mathematical Ecology. Cambridge University Press. p. 144. ISBN 0-521-00150-1.

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+{{Short description|Community Matrix}}
-In [[mathematical biology]], the '''community matrix''' is the [[linearization]] of the [[Lotka–Volterra equation]] at an [[equilibrium point]]. The [[eigenvalue]]s of the community matrix determine the stability of the equilibrium point.
+{{Use dmy dates|date=June 2016}}
+In [[mathematical biology]], the '''community matrix''' is the [[linearization]] of a [[generalized Lotka–Volterra equation]] at an [[equilibrium point]].<ref>{{cite journal|last1=Berlow|first1=E. L.
+| last2 = Neutel| first2=  A.-M. | last3 = Cohen| first3= J. E.| last4 =De Ruiter | first4=P. C. | last5 =Ebenman | first5= B.| last6 =Emmerson | first6=M. | last7 =Fox  | first7= J. W.| last8 = Jansen| first8= V. A. A.| last9 =Jones | first9=J. I. | last10 =Kokkoris | first10=G. D. | last11 =Logofet | first11=D. O. | last12 =McKane | first12= A. J. | last13 = Montoya | first13=J. M | last14 =Petchey | first14=  O.|title=Interaction Strengths in Food Webs: Issues and Opportunities |journal= Journal of Animal Ecology|volume=73|issue=5|pages=585–598|year=2004|doi=10.1111/j.0021-8790.2004.00833.x|jstor=3505669
+| doi-access=free}}</ref> The [[eigenvalue]]s of the community matrix determine the [[Lyapunov stability|stability]] of the equilibrium point.
-The Lotka–Volterra predator-prey model is
+For example, the [[Lotka–Volterra equations|Lotka–Volterra predator–prey model]] is
-:<math> \begin{align}
+:<math> \begin{array}{rcl}
-\frac{dx}{dt} &= x(\alpha - \beta y) \\
+\dfrac{dx}{dt} &=& x(\alpha - \beta y) \\
-\frac{dy}{dt} &= - y(\gamma - \delta  x),
+\dfrac{dy}{dt} &=& - y(\gamma - \delta x),
-\end{align} </math>
+\end{array} </math>
-where ''x''(''t'') denotes the number of predators, ''y''(''t'') the number of prey, and ''α'', ''β'', ''γ'' and ''δ'' are constants. The linearization of these differential equations at an equilibrium point (''x''*, ''y''*) has the form
+where ''x''(''t'') denotes the number of prey, ''y''(''t'') the number of predators, and ''α'', ''β'', ''γ'' and ''δ'' are constants. By the [[Hartman–Grobman theorem]] the non-linear system is [[Topological conjugacy|topologically equivalent]] to a linearization of the system about an equilibrium point (''x''*, ''y''*), which has the form
-:<math> \begin{bmatrix} \frac{du}{dt} \\ \frac{dv}{dt} \end{bmatrix} = A \begin{bmatrix} u \\ v \end{bmatrix}, </math>
+:<math> \begin{bmatrix} \frac{du}{dt} \\ \frac{dv}{dt} \end{bmatrix} = \mathbf{A} \begin{bmatrix} u \\ v \end{bmatrix}, </math>
-where ''u'' = ''x'' − ''x''* and ''v'' = ''y'' − ''y''*. The matrix ''A'' is called the community matrix. If ''A'' has an eigenvalue with positive real part then the equilibrium is unstable, but if all eigenvalues have negative real part then it is stable.
+where ''u'' = ''x'' − ''x''* and ''v'' = ''y'' − ''y''*. In mathematical biology, the [[Jacobian matrix]] <math>\mathbf{A}</math> evaluated at the equilibrium point (''x''*, ''y''*) is called the community matrix.<ref>{{cite book |title=Elements of Mathematical Ecology |first=Mark |last=Kot |publisher=Cambridge University Press |year=2001 |isbn=0-521-00150-1 |page=144 |url=https://books.google.com/books?id=7_IRlnNON7oC&pg=PA144 }}</ref> By the [[stable manifold theorem]], if one or both eigenvalues of <math>\mathbf{A}</math> have positive real part then the equilibrium is unstable, but if all eigenvalues have negative real part then it is stable.
+==See also==
+* [[Paradox of enrichment]]
 == References ==
+{{Reflist}}
 * {{Citation | last1=Murray | first1=James D. | title=Mathematical Biology I. An Introduction | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | series=Interdisciplinary Applied Mathematics | isbn=978-0-387-95223-9 | year=2002 | volume=17}}.
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 [[Category:Population ecology]]
 [[Category:Dynamical systems]]
+[[Category:Matrices]]
 {{mathapplied-stub}}
-[[uk:Матриця угрупування]]